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Results tagged with linear-algebra
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user 9924
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
7
votes
1
answer
878
views
The height of the Perron-Frobenius eigenvector
Does the height of a real symmetric matrix with non-negative entries control the height of its Perron-Frobenius eigenvector, under some reasonable definition of heights?
Just as an example of what ki …
- 22.6k
10
votes
3
answers
3k
views
The largest eigenvalue of a "hyperbolic" matrix
Given an integer $n\ge 1$, what is the largest eigenvalue $\lambda_n$ of the matrix $M_n=(m_{ij})_{1\le i,j\le n}$ with the elements $m_{ij}$ equal to $0$ or $1$ according to whether $ij>n$ or $ij\le …
- 22.6k
4
votes
Accepted
Projecting the unit cube onto subspaces
My answer concerns with the case $d=1$ only. Without loss of generality, we can focus on the subspaces, generated by a vector with all coordinates non-negative. It is easy to verify that for the subsp …
- 22.6k
2
votes
2
answers
594
views
Projecting the unit cube onto a subspace [closed]
I have some (rather exotic) subspace $L<R^n$, and I want to show that every non-zero vector in $\{0,1\}^n$ has a relatively small projection onto $L$. What general results and tools can be helpful? An …
- 22.6k
4
votes
The minimum rank of a matrix with a given pattern of zeros
As Misha Muzychuck has observed, the answer is "no": since
$$ \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ -1 & 0 & 1 \end{pmatrix} $$ contains a non-degenerate upper-triangular submatrix of size $2$, …
- 22.6k
2
votes
Accepted
A nowhere-zero point in a linear mapping conjecture
This is the Alon-Jaeger-Tarsi conjecture first stated in 1981 and resolved very recently (for $p\ge 83$) by János Nagy and Péter Pál Pach.
- 22.6k
5
votes
2
answers
1k
views
Is the operator norm always attained on a $\{0,1\}$-vector?
Given an operator $f\colon R^m\to R^n$, can one always find a non-zero vector
$x\in \{ 0,1 \}^m$ such that $\|f(x)\|/\|x\|\ge0.01\|f\|$? (Here I denote by
$\|\cdot\|$ both the Euclidean norms in $R^m$ …
- 22.6k
5
votes
1
answer
218
views
The minimum rank of a matrix with a given pattern of zeros
For real matrices $A=(a_{ij})$ and $B=(b_{ij})$ of the same size, I write $A\prec B$ if $a_{ij}=0$ whenever $b_{ij}=0$.
If
$$ B = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{pmatrix}^ …
- 22.6k
3
votes
Finding zero-one vectors in the row space of a matrix
This is true:
For any square matrix $M$ with all elements on its main diagonal equal to 1, and every row containing exactly two off-diagonal elements equal to −1 (with all other elements are equal to …
- 22.6k
2
votes
0
answers
103
views
Forcing zero subset sums in zero characteristic
$\newcommand{\ve}{\varepsilon}$
A problem from the year 2003 Miklos Schweitzer exam (see also this MO post) goes, essentially, as follows:
If $b_1,\dotsc,b_k\in\mathbb F_p^n$ are vectors with $b_1^\pe …
- 22.6k
2
votes
Bounding the minimal maximum norm of a solution of a linear system.
I believe you cannot give any general bound, but if the coefficients are integers, this is Siegel's Lemma: a system of $M$ equations in $N$ variables with integer coefficients $b_{ij}$ has an integer …
- 22.6k
19
votes
Accepted
Does small Perron-Frobenius eigenvalue imply small entries for integral matrices?
This is true. Indeed, you can estimate the sum of all $n^2$ elements of $A$ rather than individual elements. (Thanks to thomashennecke for observing this, my original answer dealt with the row sums of …
- 22.6k
26
votes
Accepted
Existence of a zero-sum subset
The answer is in the affirmative; indeed,
If $S$ is a finite non-empty subset of any abelian group such that every element of $S$ is a sum of two other (possibly, equal to each other) elements, then …
- 22.6k
2
votes
0
answers
78
views
A system of homogeneous linear equations
This is the "real-life" (but slightly more technical) version of a question I have asked recently.
For a prime $p>10$, let $\mathcal L_X$, $\mathcal L_Y$, and $\mathcal L_Z$ denote the pencils of al …
- 22.6k
8
votes
Accepted
Spectral radius of a proper subgraph
Here is a simple proof. Without loss of generality, $G'$ is obtained from $G$
by deleting some edges (and keeping all vertices). Let $A$ and $A'$ denote
the adjacency matrices of $G$ and $G'$, respect …
- 22.6k